Consider a continuous-time LTI system for which the input x(t) and output y(t) are related by the differential equation
(d^2 y(t) / d(t)^2) - ( dy(t)/dt) - 2y(t) = X (t).
Let X(s) and Y(s) denote Laplace transforms of x(t) and y(t), respectively, and let H(s) denote the Laplace transform of h(t), the system impulse response. (
a) Determine H(s) as a ratio of two polynomials in s domain. Sketch the pole-zero pattern of H(s).
(b) Determine h(t) for each of the following cases:
1. The system is stable.
2. The system is causal.
3. The system is neither stable nor causal.